Which Expressions Are Equivalent To 2 5 ⋅ 2 4 2^5 \cdot 2^4 2 5 ⋅ 2 4 ? Check All That Apply.A. 2 9 2^9 2 9 B. 2 20 2^{20} 2 20 C. 2 ⋅ 2 9 2 \cdot 2^9 2 ⋅ 2 9 D. 2 10 ⋅ 2 2 2^{10} \cdot 2^2 2 10 ⋅ 2 2 E. 2 − 2 ⋅ 2 11 2^{-2} \cdot 2^{11} 2 − 2 ⋅ 2 11 F. $(2 \cdot 2 \cdot 2 \cdot 2

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of equivalent expressions, specifically focusing on the expression $2^5 \cdot 2^4$. We will examine each of the given options and determine which ones are equivalent to the original expression.

The Concept of Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^5$ can be read as "2 to the power of 5" and is equivalent to $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$. This means that 2 is multiplied by itself 5 times.

The Original Expression

The original expression is $2^5 \cdot 2^4$. Using the concept of exponents, we can rewrite this expression as:

$2^5 \cdot 2^4 = 2^{5+4} = 2^9$

This is because when we multiply two numbers with the same base (in this case, 2), we can add their exponents.

Option A: $2^9$

Option A is $2^9$. As we have already shown, this is equivalent to the original expression $2^5 \cdot 2^4$. Therefore, option A is correct.

Option B: $2^{20}$

Option B is $2^{20}$. This expression is not equivalent to the original expression $2^5 \cdot 2^4$. To see why, let's rewrite the original expression as $2^9$, as we did earlier. Now, we can compare this to option B:

$2^9 \neq 2^{20}$

Since the exponents are different, option B is not equivalent to the original expression.

Option C: $2 \cdot 2^9$

Option C is $2 \cdot 2^9$. This expression can be rewritten as:

$2 \cdot 2^9 = 2^{1+9} = 2^{10}$

This is not equivalent to the original expression $2^5 \cdot 2^4$. To see why, let's compare the two expressions:

$2^5 \cdot 2^4 \neq 2^{10}$

Since the exponents are different, option C is not equivalent to the original expression.

Option D: $2^{10} \cdot 2^2$

Option D is $2^{10} \cdot 2^2$. This expression can be rewritten as:

$2^{10} \cdot 2^2 = 2^{10+2} = 2^{12}$

This is not equivalent to the original expression $2^5 \cdot 2^4$. To see why, let's compare the two expressions:

$2^5 \cdot 2^4 \neq 2^{12}$

Since the exponents are different, option D is not equivalent to the original expression.

Option E: $2^{-2} \cdot 2^{11}$

Option E is $2^{-2} \cdot 2^{11}$. This expression can be rewritten as:

$2^{-2} \cdot 2^{11} = 2^{-2+11} = 2^9$

This is equivalent to the original expression $2^5 \cdot 2^4$. Therefore, option E is correct.

Option F: $(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2)$

Option F is $(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2)$. This expression can be rewritten as:

$(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) = 2^5 \cdot 2^4 = 2^9$

This is equivalent to the original expression $2^5 \cdot 2^4$. Therefore, option F is correct.

Conclusion

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Introduction

In our previous article, we explored the concept of equivalent expressions, specifically focusing on the expression $2^5 \cdot 2^4$. We examined each of the given options and determined which ones are equivalent to the original expression. In this article, we will provide a Q&A guide to help you better understand exponents and equivalent expressions.

Q: What is an exponent?

A: An exponent is a shorthand way of representing repeated multiplication of a number. For example, $2^5$ can be read as "2 to the power of 5" and is equivalent to $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$.

Q: How do I add exponents when multiplying numbers with the same base?

A: When multiplying numbers with the same base, you can add their exponents. For example, $2^5 \cdot 2^4 = 2^{5+4} = 2^9$.

Q: What is the rule for multiplying numbers with different bases?

A: When multiplying numbers with different bases, you cannot add their exponents. For example, $2^5 \cdot 3^4$ is not equal to $2^{5+4} = 2^9$.

Q: How do I rewrite an expression with exponents in a simpler form?

A: You can rewrite an expression with exponents in a simpler form by using the rule for adding exponents. For example, $2^5 \cdot 2^4 = 2^{5+4} = 2^9$.

Q: What is the difference between $2^5$ and $2^{1+4}$?

A: $2^5$ is equal to $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, while $2^{1+4}$ is equal to $2 \cdot 2^4$. These two expressions are not equal.

Q: Can I rewrite $2^5 \cdot 2^4$ as $2^{10} \cdot 2^2$?

A: No, you cannot rewrite $2^5 \cdot 2^4$ as $2^{10} \cdot 2^2$. This is because the exponents are different.

Q: What is the correct way to rewrite $2^5 \cdot 2^4$?

A: The correct way to rewrite $2^5 \cdot 2^4$ is $2^{5+4} = 2^9$.

Q: Can I rewrite $2^5 \cdot 2^4$ as $2 \cdot 2^9$?

A: No, you cannot rewrite $2^5 \cdot 2^4$ as $2 \cdot 2^9$. This is because the exponents are different.

Q: What is the correct way to rewrite $2^5 \cdot 2^4$?

A: The correct way to rewrite $2^5 \cdot 2^4$ is $2^{5+4} = 2^9$.

Q: Can I rewrite $2^5 \cdot 2^4$ as $2^{-2} \cdot 2^{11}$?

A: Yes, you can rewrite $2^5 \cdot 2^4$ as $2^{-2} \cdot 2^{11}$. This is because $2^{-2} \cdot 2^{11} = 2^{-2+11} = 2^9$.

Q: What is the correct way to rewrite $2^5 \cdot 2^4$?

A: The correct way to rewrite $2^5 \cdot 2^4$ is $2^{5+4} = 2^9$.

Conclusion

In conclusion, we have provided a Q&A guide to help you better understand exponents and equivalent expressions. We have covered topics such as adding exponents, rewriting expressions, and multiplying numbers with different bases. We hope this guide has been helpful in clarifying any confusion you may have had about exponents and equivalent expressions.